Entanglement minimization in hadronic scattering with pions

As the initial state in the scattering process is a product state of two pions, each in the ${\bf 3}$-dimensional ($I=1$) irrep of $SU(2)$ isospin, it is convenient to work in the direct-product matrix basis. The pion isospin matrices are the three-by-three matrices $\hat{t}_{\alpha}$ which satisfy

$\displaystyle{[\,\hat{t}_{\alpha}\,,\,\hat{t}_{\beta}\,]}\,=\,i\,\epsilon_{\alpha\beta\gamma}\,\hat{t}_{\gamma}\ .$

(11)

In the product Hilbert space ${\cal H}_{1}\otimes{\cal H}_{2}$, the total isospin of the two-pion system is $\hat{\bm{t}}_{1}\otimes{\cal I}_{3}+{\cal I}_{3}\otimes\hat{\bm{t}}_{2}$, where ${\cal I}_{3}$ is the three-by-three unit matrix, which implies

$\displaystyle\hat{\bm{t}}_{1}\cdot\hat{\bm{t}}_{2}$

$\displaystyle=$

$\displaystyle\textstyle{1\over 2}\Big{[}I\left(I+1\right)\;-\;4\Big{]}\hat{\bf 1}\ =\ \hat{\bf 1}\begin{cases}-2,\qquad I=0\\ -1,\qquad I=1\\ \ \;1,\qquad I=2\end{cases}$

(12)

where $\hat{\bf 1}=\hat{\cal I}_{3}\otimes\hat{\cal I}_{3}$ and $\hat{\bm{t}}_{1}\cdot\hat{\bm{t}}_{2}=\sum\limits_{\alpha=1}^{3}\ \hat{t}_{1}^{\alpha}\otimes\hat{t}_{2}^{\alpha}$. The $9\times 9$ dimensionality of the matrix is in correspondence with the dimensionality of the $SU(2)$ isospin product representation ${\bf 3}\otimes{\bf 3}={\bf 1}\oplus{\bf 3}\oplus{\bf 5}$. There are now three invariants and three observables; one easily finds the $S$-matrix in the direct-product matrix basis

$\displaystyle\hat{\bf S}_{\ell}$

$\displaystyle=$

$\displaystyle e^{2i\delta_{\ell}^{0}}\hat{\bf P}_{0}+e^{2i\delta_{\ell}^{1}}\hat{\bf P}_{1}+e^{2i\delta_{\ell}^{2}}\hat{\bf P}_{2}\ ,$

(13)

where the three $9\times 9$ projection matrices are

	$\displaystyle\hat{\bf P}_{0}$	$\displaystyle=$	$\displaystyle-\frac{1}{3}\left(\hat{\bf 1}-\left(\hat{\bm{t}}_{1}\cdot\hat{\bm{t}}_{2}\right)^{2}\right)\ ,$		(14)
	$\displaystyle\hat{\bf P}_{1}$	$\displaystyle=$	$\displaystyle\hat{\bf 1}-\frac{1}{2}\left(\left(\hat{\bm{t}}_{1}\cdot\hat{\bm{t}}_{2}\right)+\left(\hat{\bm{t}}_{1}\cdot\hat{\bm{t}}_{2}\right)^{2}\right)\ ,$		(15)
	$\displaystyle\hat{\bf P}_{2}$	$\displaystyle=$	$\displaystyle\frac{1}{3}\left(\hat{\bf 1}+\frac{3}{2}\left(\hat{\bm{t}}_{1}\cdot\hat{\bm{t}}_{2}\right)+\frac{1}{2}\left(\hat{\bm{t}}_{1}\cdot\hat{\bm{t}}_{2}\right)^{2}\right)\ .$		(16)

It is readily checked that the $S$-matrix is unitary, and using the representation $({t}_{\gamma})_{\alpha\beta}=-i\epsilon_{\alpha\beta\gamma}$, it is straightforward to establish equivalence with the component form, Eq. (9). The trace is given by $e^{i2\delta_{\ell}^{0}}+3e^{i2\delta_{\ell}^{1}}+5e^{i2\delta_{\ell}^{2}}$ which correctly counts the isospin multiplicity, and is in correspondence with the nine eigenvalues of $\hat{\bf S}$.

A state-independent measure of the entanglement generated by the action of the $S$-matrix on the initial product state of two free particles is the EP PhysRevA.63.040304 ; mahdavi2011cross ; Beane:2018oxh . In order to compute the EP an arbitrary initial product state should be expressed in a general way that allows averaging over a given probability distribution folded with the initial state. Recall that in the $NN$ case, there are two spin states (a qubit) for each nucleon and therefore the most general initial nucleon state involves two complex parameters or four real parameters. Normalization gets rid of one parameter and there is an overall irrelevant phase which finally leaves two real parameters which parameterize the ${\bf CP}^{1}$ manifold, also known as the 2-sphere ${\bf S}^{2}$, or the Bloch sphere. Now in the isospin-one case we have three isospin states (a qutrit) which involves three complex parameters. Again normalization and removal of the overall phase reduce this to four real parameters which parameterize the ${\bf CP}^{2}$ manifold Brody_2001 ; Bengtsson:2001yd ; bengtsson_zyczkowski_2006 . Since the $\pi\pi$ initial state is the product of two isospin-one states, there will be eight parameters to integrate over to get the EP.

There are now two qutrits in the initial state, which live in the Hilbert spaces ${\cal H}_{1,2}$, each spanned by the states $\{|\,{\bf-1}_{i}\,\rangle,|\,{\bf 0}_{i}\,\rangle,|\,{\bf 1}_{i}\,\rangle\}$ with $i=1,2$. It is of interest to determine the EP of a given $S$-matrix operator, which is a measure of the entanglement of the scattered state averaged over the ${\bf CP}^{2}$ manifolds on which the qutrits live. Consider an arbitrary initial product state of the qutrits

$\displaystyle|\,\Psi\,\rangle\ =\ U\left(\alpha_{1},\beta_{1},\mu_{1},\nu_{1}\right)|\,\rangle_{1}\otimes U\left(\alpha_{2},\beta_{2},\mu_{2},\nu_{2}\right)|\,\rangle_{2}$

(20)

with

$\displaystyle U\left(\alpha_{i},\beta_{i},\mu_{i},\nu_{i}\right)|\,\rangle_{i}=\cos\beta_{i}\sin\alpha_{i}|\,{\bf-1}\,\rangle_{i}+e^{i\mu_{i}}\sin\beta_{i}\sin\alpha_{i}|\,{\bf 0}\,\rangle_{i}+e^{i\nu_{i}}\cos\alpha_{i}|\,{\bf 1}\,\rangle_{i}\,,$

(21)

where $0\leq{\mu_{i},\nu_{i}}<2{\pi}$ and $0\leq{\alpha_{i},\beta_{i}}\leq{\pi}/2$. The geometry of ${\bf CP}^{2}$ is described by the Fubini-Study (FS) line element Brody_2001 ; Bengtsson:2001yd ; bengtsson_zyczkowski_2006

$\displaystyle\begin{split}ds_{\scriptstyle FS}^{2}=\;&d\alpha^{2}+\sin^{2}(\alpha)d\beta^{2}+\left(\sin^{2}(\alpha)\sin^{2}(\beta)-\sin^{4}(\alpha)\sin^{4}(\beta)\right)d\mu^{2}+\\ &\sin^{2}(\alpha)\cos^{2}(\alpha)d\nu^{2}-2\sin^{2}(\alpha)\cos^{2}(\alpha)\sin^{2}(\beta)d\mu d\nu\ .\end{split}$

(22)

Of special interest here is the differential volume element which in these coordinates is

	$\displaystyle dV_{\scriptstyle FS}$	$\displaystyle=$	$\displaystyle\sqrt{det\,g_{\scriptstyle FS}}\,d\alpha\,d\beta\,d\mu\,d\nu$		(23)
		$\displaystyle=$	$\displaystyle\cos\alpha\cos\beta\sin^{3}\alpha\sin\beta\,d\alpha\,d\beta\,d\mu\,d\nu\$

and the volume of the ${\bf CP}^{2}$ manifold is found to be,

$\displaystyle\int dV_{\scriptstyle FS}$

$\displaystyle=$

$\displaystyle\frac{\pi^{2}}{2}\ .$

(24)

The final state of the scattering process is obtained by acting with the unitary $S$-matrix of definite angular momentum on the arbitrary initial product state:

$\displaystyle|\,\bar{\Psi}\,\rangle\ =\ \hat{\bf S}_{\ell}|\,\Psi\,\rangle\ .$

(25)

The associated density matrix is

$\displaystyle\rho_{1,2}\ =\ |\,\bar{\Psi}\,\rangle\langle\,\bar{\Psi}|\,,$

(26)

and tracing over the states in ${\cal H}_{2}$ gives the reduced density matrix

$\displaystyle\rho_{1}\ =\ {\rm Tr}_{2}\big{[}\rho_{1,2}\big{]}.$

(27)

The linear entropy of the $S$-matrix is then defined as⁵⁵5Note that this is related to the (exponential of the) second Rényi entropy.

$\displaystyle E_{\hat{\bf S}_{\ell}}\ =\ 1\ -\ {\rm Tr}_{1}\big{[}\left(\rho_{1}\right)^{2}\big{]}.$

(28)

Finally, the linear entropy is integrated over the initial ${\bf CP}^{2}$ manifolds to form the average, and the entanglement power is

$\displaystyle{\mathcal{E}}({\hat{\bf S}_{\ell}})\ =\ \left(\frac{2}{\pi^{2}}\right)^{2}\left(\prod_{i=1}^{2}\int dV^{i}_{\scriptstyle FS}\right){\cal P}E_{\hat{\bf S}_{\ell}}$

(29)

where ${\cal P}$ is a probability distribution which here will be taken to be unity. Evaluating this expression using Eq. (13) yields the s-wave $\pi\pi$ EP:

	$\displaystyle{\mathcal{E}}({\hat{\bf S}_{0}})$	$\displaystyle=$	$\displaystyle\frac{1}{648}\left(156-6\cos[4\delta_{0}^{0}]-65\cos[2(\delta_{0}^{0}-\delta_{0}^{2})]\right.$		(30)
			$\displaystyle\qquad\qquad\qquad\left.-10\cos[4(\delta_{0}^{0}-\delta_{0}^{2})]-60\cos[4\delta_{0}^{2}]-15\cos[2(\delta_{0}^{0}+\delta_{0}^{2})]\right)\ ,$

and the p-wave $\pi\pi$ EP:

$\displaystyle{\mathcal{E}}({\hat{\bf S}_{1}})=\frac{1}{4}\sin^{2}\left(2\delta_{1}^{1}\right)\ .$

(31)

Notice that this matches the intuitive construction which led to Eq. (19). The EPs have the non-entangling solutions:

	$\displaystyle\delta_{0}^{0}$	$\displaystyle=$	$\displaystyle\delta_{0}^{2}\ =\ 0,\frac{\pi}{2}\ ,$		(32)
	$\displaystyle\delta_{1}^{1}$	$\displaystyle=$	$\displaystyle 0,\frac{\pi}{2}\ .$		(33)

Therefore, in the s-wave, entanglement minimization implies that both isospins are either non-interacting or at resonance, while in the p-wave, entanglement minimization implies that the $I=1$ channel is either non-interacting or at resonance. As no $I=2$ resonances are observed in nature (and their coupling to pions is suppressed in large-${N_{c}}$ QCD Weinberg:2013cfa ), the s-wave EP has a single minimum corresponding to no interaction. By contrast, the $I=1$ channel will exhibit minima of both types. It is worth considering the EP of a simple resonance model. Consider the unitary $S$-matrix:

$\displaystyle{\hat{\bf S}_{1}}\ =\ \frac{s-m_{1}^{2}-im_{1}\Gamma_{1}}{s-m_{1}^{2}+im_{1}\Gamma_{1}}\ ,$

(34)

where $m_{1}$ ($\Gamma_{1}$) are the mass (width) of the resonance. The EP is

$\displaystyle{\mathcal{E}}({\hat{\bf S}_{1}})=\left(\frac{m_{1}\Gamma_{1}\left(s-m_{1}^{2}\right)}{\left(m_{1}\Gamma_{1}\right)^{2}+\left(s-m_{1}^{2}\right)^{2}}\right)^{2}\ ,$

(35)

which vanishes on resonance at $s=m_{1}^{2}$ and has maxima at $s=m_{1}(m_{1}\pm\Gamma_{1})$. It is clear that the minimum corresponds to $\hat{\bf S}\propto{\cal P}_{12}$. As the $\rho$-resonance dominates the $I=1$ channel at energies below $1~{}{\rm GeV}$, the EP in nature will be approximately of this form.

The $\pi\pi$ phase shifts are the most accurately known of all hadronic $S$-matrices as the Roy equation constraints Roy:1971tc come very close to a complete determination of the phase shifts Ananthanarayan:2000ht ; Colangelo:2001df . In Fig. (1) the EPs for the first few partial waves are plotted using the Roy equation determinations of the $S$-matrix.

The $S$-matrix element for the scattering process, $\pi^{a}(q_{1})N(p_{1})\to\pi^{b}(q_{2})N(p_{2})$, is given by

$$\langle\pi^{b}(q_{2})N(p_{2})|S|\pi^{a}(q_{1})N(p_{1})\rangle=\langle\pi^{b}(q_{2})N(p_{2})|\pi^{a}(q_{1})N(p_{1})\rangle\\ +\langle\pi^{b}(q_{2})N(p_{2})|iT|\pi^{a}(q_{1})N(p_{1})\rangle,$$

(39)

where $a$ and $b$ label the isospin of the pion. The $T$ matrix element in the center-of-mass system (cms) for the process may be parameterized as Fettes_1998

$$\begin{split}T^{ba}_{\pi N}=\left({E+m\over 2m}\right)\bigg{\{}\delta^{ba}&\left[g^{+}(\omega,t)+i\vec{\sigma}\cdot(\vec{q_{2}}\times\vec{q_{1}})h^{+}(\omega,t)\right]\\ +i\epsilon^{abc}\tau^{c}&\left[g^{-}(\omega,t)+i\vec{\sigma}\cdot(\vec{q_{2}}\times\vec{q_{1}})h^{-}(\omega,t)\right]\bigg{\}}\end{split}$$

(40)

where $E$ is the nucleon energy, $\omega$ is the pion energy, $m$ is the nucleon mass and $t=(q_{1}-q_{2})^{2}$ is the square of the momentum transfer. The $\sigma$($\tau$) matrices act on the spin(isospin) of the incoming nucleon. This decomposition reduces the scattering problem to calculating $g^{\pm}$, the isoscalar/isovector non-spin-flip amplitude and $h^{\pm}$, the isoscalar/isovector spin-flip amplitude. The amplitude can be further projected onto partial waves by integrating against $P_{\ell}$, the relevant Legendre polynomial:

$$f_{\ell\pm}^{\pm}(s)={E+m\over 16\pi\sqrt{s}}\int_{-1}^{+1}dz\left[g^{\pm}P_{\ell}(z)+{\vec{q}}^{\,2}h^{\pm}\big{(}P_{\ell\pm 1}(z)-zP_{\ell}(z)\big{)}\right]\ .$$

(41)

Here $z=\cos{\theta}$ is the cosine of the scattering angle, $s$ is the cms energy squared and $\vec{q}^{\,2}=\vec{q_{1}}^{2}=\vec{q_{2}}^{2}$. The subscript $\pm$ on the partial wave amplitude indicates the total angular momentum $J=\ell\pm s$. The amplitudes in the total isospin $I=\frac{1}{2}$ and $I=\frac{3}{2}$ can be recovered via the identification:

$$f_{\ell\pm}^{\frac{1}{2}}=f_{\ell\pm}^{+}+2f_{\ell\pm}^{-}\ \ ,\ \ f_{\ell\pm}^{\frac{3}{2}}=f_{\ell\pm}^{+}-f_{\ell\pm}^{-}\ .$$

(42)

Below inelastic threshold the scattering amplitude is related to a unitary $S$-matrix by

$$S_{\ell\pm}^{I}(s)=1+2i\lvert\,\vec{q}\,\rvert f_{\ell\pm}^{I}(s)\ \ ,\ \ S_{\ell\pm}^{I}(s)S_{\ell\pm}^{I}(s)^{\dagger}=1$$

(43)

and the $S$-matrix can be parameterized in terms of phase shifts,

$$S_{\ell\pm}^{I}(s)=e^{2i\delta_{\ell\pm}^{I}(s)}\ .$$

(44)

For a more detailed derivation of the $\pi N$ $S$-matrix see Fettes_1998 ; Yao_2016 ; Moj_i__1998 ; Scherer2012 . Scattering in a given partial wave and total angular momentum channel leads to a $S$-matrix which acts on the product Hilbert space of the nucleon and pion isospin, $\mathcal{H}_{\pi}\otimes\mathcal{H}_{N}$. The $S$-matrix can then be written in terms of total isospin projection operators

$$\hat{{\bf S}}_{\ell\pm}=e^{2i\delta_{\ell\pm}^{3/2}}\hat{{\bf P}}_{3/2}+e^{2i\delta_{\ell\pm}^{1/2}}\hat{{\bf P}}_{1/2}$$

(45)

where the $6\times 6$ projection matrices are

$$\begin{split}&\hat{{\bf P}}_{3/2}=\frac{2}{3}\left(\hat{{\bf 1}}+\hat{{\bf t}}_{\pi}\cdot\hat{{\bf t}}_{N}\right)\ ,\\ &\hat{{\bf P}}_{1/2}=\frac{1}{3}\left(\hat{{\bf 1}}-2(\hat{{\bf t}}_{\pi}\cdot\hat{{\bf t}}_{N})\right)\ .\end{split}$$

(46)

The operators $\hat{{\bf t}}_{N}$ and $\hat{{\bf t}}_{\pi}$ are in the $2$ and $3$ dimensional representations of $SU(2)$ isospin respectively and $\hat{\bm{t}}_{\pi}\cdot\hat{\bm{t}}_{N}=\sum\limits_{\alpha=1}^{3}\ \hat{t}_{\pi}^{\alpha}\otimes\hat{t}_{N}^{\alpha}$.

The entanglement power of the $\pi N$ $S$-matrix can be computed in a similar manner as for the $\pi\pi$ EP. The incoming separable state now maps to a point on the product manifold, ${\bf CP}^{2}\times{\bf S}^{2}$. The construction of the reduced density matrix follows the same steps as in section 2.2 and the entanglement power is found to be,

$$\begin{split}{\mathcal{E}}({\hat{\bf S}}_{\ell\pm})&=\left(\frac{2}{\pi^{2}}\frac{1}{4\pi}\right)\left(\int dV_{FS}d\Omega\right)\mathcal{P}E_{\hat{\bf S}_{\ell\pm}}\\ &=\frac{8}{243}\bigg{[}17+10\cos\left(2\big{(}\delta_{\ell\pm}^{{3/2}}-\delta_{\ell\pm}^{{1/2}}\big{)}\right)\bigg{]}\sin^{2}\left(\delta_{\ell\pm}^{{3/2}}-\delta_{\ell\pm}^{{1/2}}\right)\end{split}$$

(47)

where $\mathcal{P}$ has been taken to be $1$. Note that the two particles are now distinguishable and so scattering in each partial wave is no longer constrained by Bose/Fermi statistics. It follows that the $S$-matrix is only non-entangling when it is proportional to the identity which occurs when,

$$\delta_{\ell\pm}^{{3/2}}=\delta_{\ell\pm}^{{1/2}}\ .$$

(48)

Notice that the EP allows for interesting local minima when the difference in $I=3/2$ and $I=1/2$ phase shifts is $\pi/2$. The $\pi N$ phase shifts are determined very accurately by the Roy-Steiner equations up to a center-of-mass energy of 1.38 GeV Hoferichter_2016 and the entanglement power for the first couple partial waves is shown in Fig. (2).

There is a local minimum near the delta resonance position in the p-wave due to the rapid change of the $I=3/2$ phase shift.

Near threshold the phase shifts can be determined by the scattering lengths through the effective range expansion,

$$\delta_{\ell\pm}^{I}=\cot^{-1}{\left\{{1\over\lvert\,\vec{q}\,\rvert^{2\ell+1}}\left({1\over a_{\ell\pm}^{I}}+\mathcal{O}(\vec{q}^{\,2})\right)\right\}}\ .$$

(49)

This leads to the threshold form of the entanglement power,

$${\mathcal{E}}({\hat{\bf S}}_{\ell\pm})=\frac{8}{9}\left(a^{1\over 2}_{\ell\pm}-a^{3\over 2}_{\ell\pm}\right)^{2}\vec{q}^{\,2+4l}$$

(50)

which can only vanish if $a^{1\over 2}_{\ell\pm}=a^{3\over 2}_{\ell\pm}$. The scattering lengths at leading order in heavy-baryon chiral perturbation theory, including the delta, are given by Fettes_1998 ; Fettes_2001 ,

$$\begin{split}&a^{1\over 2}_{0+}=\frac{2M_{\pi}m}{8\pi(m+{M_{\pi}})F_{\pi}^{2}}\ \ ,\ \ a^{3\over 2}_{0+}=\frac{-M_{\pi}m}{8\pi(m+{M_{\pi}})F_{\pi}^{2}}\\ &a^{1\over 2}_{1-}=-\frac{m\left(9g_{A}^{2}\Delta+9g_{A}^{2}M_{\pi}-8g_{\pi N\Delta}^{2}M_{\pi}\right)}{54\pi F_{\pi}^{2}M_{\pi}(\Delta+M_{\pi})(m+M_{\pi})}\ \ ,\ \ a^{3\over 2}_{1-}=-\frac{m\left(9g_{A}^{2}\Delta+9g_{A}^{2}M_{\pi}-8g_{\pi N\Delta}^{2}M_{\pi}\right)}{216\pi F_{\pi}^{2}M_{\pi}(\Delta+M_{\pi})(m+M_{\pi})}\\ &a^{1\over 2}_{1+}=\frac{m\left(-3g_{A}^{2}\Delta+3g_{A}^{2}M_{\pi}+8g_{\pi N\Delta}^{2}M_{\pi}\right)}{72\pi F_{\pi}^{2}M_{\pi}(\Delta-M_{\pi})(m+M_{\pi})}\ \ ,\ \ a^{3\over 2}_{1+}=\frac{m\left(-3g_{A}^{2}\Delta^{2}-2g_{\pi N\Delta}^{2}M_{\pi}\Delta+3g_{A}^{2}M_{\pi}^{2}\right)}{36\pi F_{\pi}^{2}M_{\pi}\left(M_{\pi}^{2}-\Delta^{2}\right)(m+M_{\pi})}\end{split}$$

(51)

where $\Delta=m_{\Delta}-m_{N}$ is the delta-nucleon mass splitting. The corresponding EPs near threshold are,

$$\begin{split}&{\mathcal{E}}({\hat{\bf S}}_{0+})=\frac{m^{2}M_{\pi}^{2}}{8\pi^{2}F_{\pi}^{4}(m+M_{\pi})^{2}}\vec{q}^{\,2}\\ &{\mathcal{E}}({\hat{\bf S}}_{1-})=\frac{m^{2}\left(9g_{A}^{2}\Delta+9g_{A}^{2}M_{\pi}-8g_{\pi N\Delta}^{2}M_{\pi}\right)^{2}}{5832\pi^{2}f_{\pi}^{4}M_{\pi}^{2}(\Delta+M_{\pi})^{2}(m+M_{\pi})^{2}}\vec{q}^{\,6}\\ &{\mathcal{E}}({\hat{\bf S}}_{1+})=\frac{m^{2}\left(-9g_{A}^{2}\Delta^{2}+4g_{\pi N\Delta}^{2}\Delta M_{\pi}+\left(9g_{A}^{2}+8g_{\pi N\Delta}^{2}\right)M_{\pi}^{2}\right)^{2}}{5832\pi^{2}f_{\pi}^{4}M_{\pi}^{2}\left(M_{\pi}^{2}-\Delta^{2}\right)^{2}(m+M_{\pi})^{2}}\vec{q}^{\,6}\ .\end{split}$$

(52)

Once again the only non-entangling solution consistent with chiral symmetry is no interaction, with the same scaling of $F_{\pi}$ as found in $\pi\pi$ scattering. Unlike the large-${N_{c}}$ limit, there is no reason to expect an enhancement of the axial couplings, which in that case gives rise to the contracted spin-flavor symmetries Dashen:1993as .